Source:* **Journal of Mathematics Teacher Education, 14(1),* p. 25–47. ( February, 2011).

(Reviewed by the Portal Team)

In this article, the authors report on a conceptual framework developed for identifying and analyzing mathematical features of classroom work.

The authors describe their method, including how the authors synthesized the literature on mathematics instruction in classrooms and how they developed their coding scheme.

Next, the authors share their conceptualization of the mathematical quality of instruction (MQI) by providing coding guidelines for particular constructs and by illustrating the application of specific codes to two example lessons.

In this conceptualization, the authors' focus is more on mathematical features closely related to the work of teaching, such as the presence of error and how it is treated, or precision in the use of mathematical language.

**Method **The authors decided to make video recordings of classroom instruction for later description and coding.

Following this decision, the authors recruited 10 teachers based on their participation in select professional development workshops.

Teachers taught various grades from second to sixth, although the sixth-grade teacher moved to eighth grade in the second year of the study.

Seven taught in districts serving families from a wide range of social, economic, and cultural backgrounds, including many non-native English speakers.

The three other teachers taught in one school in a small, upper-class, primarily Caucasian district.

The authors collected three forms of data:

1. Videotaped lessons - classrooms were taped for a total of nine lessons.

2. Paper-and-pencil surveys - teachers each completed a survey designed to assess their mathematical knowledge for teaching (Hill et al. 2004).

3. Post-lesson interviews - in the post-lesson interviews (one for each videotaped lesson), the teachers were asked to clarify their lesson goals and the extent to which they believed these goals had been met. They were also asked to identify mathematical (or other) difficulties they encountered and/or struggles they perceived their students had during the lesson.

The authors' efforts to describe the MQI resulted in a set of constructs and codes which can be used to capture key mathematical events in classrooms.

These constructs tend to fall into two categories: negative instances of classroom mathematics (e.g., errors, responding to students inappropriately) and positive instances of classroom mathematics (e.g., rich mathematics, the proper use of mathematical language, promoting equity).

These two categories reflect findings from the literature about how teachers’ mathematical knowledge affects classroom practice (Hill et al. 2008).

The authors argue that their theorization of this domain is a good beginning and brings together what have until now been disparate perspectives about what matters, mathematically, in classrooms.

Though the two observational systems measure different constructs and have different purposes, they converge around some of their key classroom foci: teachers’ facility in responding to students, making connections, and choosing and using examples and representations.

Reference

Hill, H. C., Blunk, M., Charalambous, C., Lewis, J., Phelps, G. C., Sleep, L., et al. (2008). Mathematical knowledge for teaching and the mathematical quality of instruction: An exploratory study. *Cognition and Instruction*, 26(4), 430–511.

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